Aspects of Wrapped Branes in String and M-Theory Ling Bao

Transcription

1 Thesis for the degree of Doctor of Philosophy Aspects of Wrapped Branes in String and M-Theory Ling Bao Department of Fundamental Physics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2009

2 Aspects of Wrapped Branes in String and M-Theory Ling Bao ISBN Ling Bao, Doktorsavhandlingar vid Chalmers tekniska högskola Ny serie nr 3001 ISSN X Department of Fundamental Physics Chalmers University of Technology SE Göteborg, Sweden Telephone +46-(0) Chalmers Reproservice Göteborg, Sweden 2009

3 Aspects of Wrapped Branes in String and M-Theory Ling Bao Department of Fundamental Physics Chalmers University of Technology SE Göteborg, Sweden Abstract This thesis consists of an introductory text together with five appended research papers. The Ariadne s thread through the whole thesis is various effects coming from high-dimensional p-branes in various subsectors of string and M-theory. The low energy effective actions in string and M-theory consists of a classical supergravity together with quantum corrections. In particular the nonperturbative correction terms arise from instanton effects, which are interpreted as p-branes wrapping supersymmetric cycles. The general structure of the full effective action is the result of a complicated interplay between supersymmetry and U-duality. Requiring the action to be invariant under U-duality leads to mathematical functions called automorphic forms. Both perturbative and non-perturbative corrections seem to be captured by these functions. The U-duality groups can be found by analyzing the algebraic structures of the moduli space after toroidal compactification. Using this line of thinking, some simple examples of higher order derivative corrections in pure gravity are investigated. Compactification on manifolds with special holonomy is also discussed in this thesis, with focus on the resulting moduli spaces. Certain quantum corrections to type IIA string theory compactified on a rigid Calabi-Yau threefold are analyzed. Manifolds with special holonomy constitute target spaces of the topological subsectors in string and M-theory. The low energy effective action of these theories consists of a classical contribution from a form theory of gravity, which receives quantum corrections from branes wrapping supersymmetric cycles in the target space. In particular the dynamics of the M2- and M5-branes are discussed in the context of a topological version of M-theory. iii

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5 This thesis consists of an introductory text and the following five appended research papers, henceforth referred to as Paper I-V: I. L. Bao, V. Bengtsson, M. Cederwall and B. E. W. Nilsson, Membranes for topological M-theory, JHEP0601 (2006) 150, [hep-th/ ]. II. L. Bao, M. Cederwall and B. E. W. Nilsson, A note on topological M5- branes and string-fivebrane duality, JHEP 0806 (2008) 100, [hep-th/ ]. III. L. Bao, M. Cederwall and B. E. W. Nilsson, Aspects of higher curvature terms and U-duality, Class. Quant. Grav. 25 (2008) , [ [hep-th]]. IV. L. Bao, J. Bielecki, M. Cederwall, B. E. W. Nilsson and D. Persson, U- Duality and the Compactified Gauss-Bonnet Term, JHEP0807 (2008) 048, [ [hep-th]]. V. L. Bao, A. Kleinschmidt, B. E. W. Nilsson, D. Persson and B. Pioline, Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1), toappear. v

6 Acknowledgments There are many people who has contributed to this thesis, one way or another, that I would like to thank. My deepest gratitude goes to my supervisor Professor Bengt E. W. Nilsson, who has been a teacher and a mentor during my years as a Ph.D. student. His way of always giving constructive advices has taught me more than just physics. I am very grateful to all the collaborators I had pleasure to work with: Viktor Bengtsson, Johan Bielecki, Martin Cederwall, Axel Kleinschmidt, Daniel Persson and Boris Pioline. Especially Professor Martin Cederwall, who has been my stand-in supervisor whenever I needed. During my time at Chalmers, I have had countless fruitful discussions from which I have learned many things. I would like to acknowledge the past and present members of the department for all these discussions and also for creating such a cozy atmosphere: Pär Arvidsson, Lars Brink, Ludde Edgren, Gabriele Ferretti, Erik Flink, Ulf Gran, Rainer Heise, Måns Henningson, Robert Marnelius, Fredrik Ohlsson, Christoffer Petersson, Per Salomonson and Niclas Wyllard. I am thankful for the assistance I have received from all the members of our subatomic sister group, and also for the help from our solve it all secretary Kate Larsson. During my Ph.D. studies I spent three stimulating months at the Albert Einstein Institute in Germany. I would like to thank AEI for the hospitality. I am very grateful to Carlos R. Mafra, Jakob Palmkvist and Daniel Persson, who gave invaluable comments on this manuscript. I would like to thank Pontus for the love and encouragement. I am also grateful to my sister, who always gives me the greatest support. Last but not least my dear mother, who bestowed me my curiosity for physics and stands behind me whatever decisions I make. vi

7 Contents Introduction 1 Outline 7 1 Supergravities and Dualities Higher-Dimensional Supergravites Eleven-Dimensional Supergravity Type IIA Supergravity Type IIB Supergravity The Democratic Formulation S-duality T-duality U-duality Web of Dualities Compactification and Geometry Torus Compactification Compactification on a Circle Generalization to n-torus Coset Symmetry Calabi-Yau Compactification Calabi-Yau Manifold Calabi-Yau Three-fold Compactification of Type II Strings Non-perturbative Instanton Effects G 2 Manifold The Topological Sector Six Dimensions Seven Dimensions String Effective Actions Scattering Amplitudes U-duality Completion vii

8 3.3 Supersymmetry Beyond Type IIB String Theory Automorphic Forms Modular Forms The Modular Group Definition of Modular Forms Towards Automorphic Forms Definition of Automorphic Forms Constructing Automorphic Forms Fourier Expansion Transforming Automorphic Forms A p-adic Numbers 83 Bibliography 85 Papers I-V 99

9 Introduction Modern physics started its course almost a century ago with the birth of quantum mechanics and general relativity. In many ways these two theories can be considered as opposite poles. History has told us that these two pillars of theoretical physics seem to be incompatible with each other. However, there are reasons to hope that by choosing a clever language this can be rectified and the two theories made to live in perfect harmony with each other. Finding the appropriate framework to do so has been the ultimate quest of high energy physics for the last three decades. Einstein s theory of general relativity couples gravitational motion to the geometry of spacetime. Gravitational systems are governed by equations of motion which retain their form under coordinate transformations. This theory works extremely well for heavy objects over large distance scales, as in astronomy for instance. Quantum mechanics, on the other hand, dictates that the energy cannot take arbitrarily small values. Rather, it is said to be quantized. Quantum mechanics is the framework to use when dealing with small physical systems like atoms. The experimental discoveries of the electromagnetic, weak and strong forces pointed towards a general picture of the fundamental constituents of Nature. Both matter and forces are viewed as point-like particles, which are characterized by mass, spin, charge, etc.. Some of them obey Bose-Einstein statistics (bosons), others follow Fermi-Dirac statistics (fermions). In particular the forces are mediated through massless particles named gauge bosons. Since the gauge bosons move at the speed of light, the correct quantum theory describing these has to respect Lorentz symmetry. In this so called relativistic quantum field theory, elementary particles appear as states in the spectrum after quantization. The particle dynamics are then dictated by the scattering amplitudes, which are derived in terms of Feynman diagrams. The construction of the Standard Model containing the electromagnetic, weak and strong forces is so far the greatest success of quantum field theory. The Standard Model is a nonabelian gauge theory based on the Lie group SU(3) SU(2) U(1). Although some of the Feynman diagrams seemed to give rise to divergences at first, it was later found that they can be canceled out by employing a clever renormalization scheme. Experimentally the Standard Model has been tested and seen 1

10 to hold extremely well. The last major piece of the puzzle still missing is the experimental verification of the mechanism that gives mass to the elementary particles, which presumably happens via the Higgs mechanism. The development of the Standard Model has been a process moving higher and higher up in energy scale, theoretically as well as experimentally. Extrapolating the three coupling constants in the theory to very high energies, it turns out that they intersect almost at one point. A new kind of symmetry which exchanges bosons with fermions then entered the stage. This supersymmetry made the Higgs sector of the Standard Model better behaved in the ultraviolet region, and as a side product the three forces became beautifully united. Hopefully the answer to whether or not supersymmetry really exists will be found in a not too distant future. The idea of uniting the forces in Nature has been an enormously fruitful guide for theoretical physics during the twentieth century. It was this line of thinking that enabled the construction of the Standard Model. However, the theory describing the origin of the universe or interior of a black hole must contain both gauge theories and gravity. In other words, there exists a length scale, known as the Planck length ħg l p = c (81) m, (1) where the spacetime itself is quantized. The ingredients in Eq. (1) are the three fundamental constants in Nature 1 : the reduced Planck constant ħ, the gravitational constant G and the speed of light in vacuum c. Straightforwardly quantizing gravity leads to many problems that we do not know how to solve, e.g., it seems to be non-renormalizable. The most successful attempt at quantizing gravity up to now is provided by string theory. The fundamental object in string theory is, as the name suggests, a string, which when moving around sweeps out a two-dimensional surface in spacetime named the worldsheet. The classical action is simply given by the area of the worldsheet. Quantizing this action, one finds not only gauge fields, but also a particle that can be interpreted as the graviton. One of the many beautiful properties of string theory is the fact that it contains only one free parameter the Regge slope α. Both the characteristic length scale l s and the tension T F1,S of the fundamental string are expressed in terms of α according to l s = α and T F1,S = 1 2πα. (2) Moreover, the coupling constant appearing in target space is identified with the vacuum expectation value of the dilaton scalar field: g s = e <φ>. (3) 1 Throughout this thesis we will use natural units, i.e., ħ = c =1. 2

11 The fact that strings are one-dimensional resolves the problem of ultraviolet divergences in the scattering amplitudes. The major argument against string theory is that it requires a huge number of spacetime dimensions to be consistent. Requiring also invariance under supersymmetry constrains the dimension to be ten, which is still way to many compared to the four we observe. The basic idea of how to deal with this problem is Kaluza-Klein compactification, where the six superfluous spacelike dimensions are thought to be small. Although we cannot observe these compact dimensions directly, the four-dimensional physics is affected by their detailed structures. Much efforts have been made trying to understand the exact implications of various choices of internal manifolds. Another puzzle in string theory for a long time was the fact that five selfconsistent string theories with seemingly distinct properties were found. They are called type IIA, type IIB, type I, O(32) heterotic and E 8 E 8 heterotic. In the low energy limit each of them is described by a corresponding supergravity theory together with perturbative as well as non-perturbative quantum corrections. This puzzle was eliminated by the discovery of dualities. S-duality exchanges weak and strong couplings, while T-duality relates certain string theories compactified on small radii with others compactified on large radii. These dualities collectively point towards an eleven-dimensional umbrella, which is named M-theory by E. Witten. For instance type IIA string theory is obtained from M-theory by circular compactification, in particular the IIA string coupling constant can be reinterpreted in terms of the compactification radius R 11 and the Planck length, ( ) 3/2 R11 g s =. (4) l p In the low energy limit, M-theory itself is described by an eleven-dimensional supergravity theory. As for the quantum theory, the fundamental object is believed to be a membrane. However, quantization of the membrane worldvolume action has so far not been achieved in a satisfactory way. Whatever the microscopic description of M-theory turns out to be, the various string theories should be thought of as perturbative descriptions of distinct corners of its parameter space. Dualities are the correct tool to use when relating these corners. Although string and M-theory are mathematically very beautiful, we shall not forget that the goal for physicists is to understand Nature. The discovery of higher-dimensional p-branes in string theory finally opened the door to semi-realistic gauge theories. Furthermore, during the last decade a new type of duality was discovered, which relates certain string configurations on some particular d-dimensional spacetime geometries to nonabelian gauge theories in one dimension less. As string and M-theory reveal more and more of their secrets, hopefully soon we will know whether or not this is the right track to take. 3

12 D-Branes A special type of p-branes related to open strings with Dirichlet boundary conditions will play a central role in this thesis. Some basic properties of these so called D-branes are briefly reviewed here. Dp-branes in string theory are (p + 1)-dimensional objects on which open strings can end. They were first discovered as a consequence of the T-duality by J. Dai, R. G. Leigh and J. Polchinski [1], and independently by P. Hořava [2]. Later they were identified with BPS p-brane solutions of the ten-dimensional supergravity theories [3]. The presence of D-branes breaks the symmetries of the Minkowski space vacuum. In the vicinity of a Dp-brane the Lorentz symmetry is broken according to SO(1, 9) SO(1,p) SO(9 p), (5) while at least half of the supersymmetries are also broken. Every massless gauge field in string theory is generated by an electric or magnetic p-brane source. Let A n denote an n-form gauge field. Its field strength is then an (n + 1)-form given by F n+1 = da n. (6) Coupling terms containing the other gauge fields are for simplicity omitted on the right hand side. Due to d 2 = 0, field strengths defined as in Eq. (6) are invariant under the gauge transformations δa n = dλ n 1. (7) A(p + 1)-form gauge field can be coupled to a p-brane via S int = e p A p+1, (8) where the pullback of A p+1 to the brane worldvolume is understood implicitly. The conventions we use are from Ref. [4]. The electric p-brane charge can be computed using Gauss s law e p = F p+2. (9) S D p 2 The integral (9) is computed over a sphere S D p 2, with D being the total number of spacetime dimensions. On the other hand, one can also define a magnetic charge according to m p = F p+2. (10) S p+2 4

13 Making the identification F p+2 = F D p 2, we may reinterpret m p as the electric charge of a dual (D p 4)-brane S int = m p A D p 3, (11) where A D p 3 is the gauge potential of FD p 2. Thus, p- and(d p 4)- branes are dual to each other. In particular for D = 10 we find that p-branes are dual to (6 p)-branes, and it is motivated to set e p 6 = m p. Moreover, the electric and magnetic charges have to satisfy the Dirac quantization condition [5, 6, 7] e p m p = e p e D p 4 2πZ. (12) The Dp-branes are most naturally embedded in a target space, containing both spacetime and (odd) Grassmann coordinates, called superspace [8]. Theories formulated in superspace are manifestly target space supersymmetric. On the worldsheet local kappa symmetry is employed to ensure the matching between bosonic and fermionic degrees of freedom [9, 10, 11, 12, 13, 14]. The worldvolume theory of a single Dp-brane in type II string theories is governed by the Dirac-Born-Infeld action [15] S p = T Dp d p+1 σ det(g αβ +2πα F αβ ). (13) Here G αβ is the worldvolume pullback of the spacetime metric, while F αβ is the pullback of a combination with the Maxwell field strength and the Kalb- Ramond field. The worldvolume coordinates are denoted by σ α. The action in Eq. (13) is non-linear, and expanding with respect to small F αβ leads to an infinite series of terms starting with the ordinary Maxwell action. The symbol T Dp stands for the tension of the p-brane. Using T-duality one can find the general expression 1 T Dp =. (14) g s (2π) p α p+1 2 Let us emphasize the fact that the tension of a D-brane behaves as the inverse of the string coupling constant, i.e., T Dp 1/g s. Later we will also encounter the so called NS5-branes, whose tension scales like T NS5 1/gs 2. D-branes play an important role in string theory, since gauge theories arise naturally on the D-brane worldvolume [16]. This gives rise to new opportunities to find the Standard Model. From the gravitational point of view, D-branes living entirely in the compact dimensions provide a microscopic explanation for the thermodynamical properties of black holes [17]. In this thesis we will focus on D-branes wrapping supersymmetric cycles in a compact manifold. Two examples of such D-brane effects will be given. One is excitations 5

14 of topological string theories on Calabi-Yau threefolds. The other is instanton corrections in the spacetime effective theory, arising when D-branes are completely wrapped on cycles in the internal manifold. 6

15 Outline This thesis consists of four chapters, one appendix and five research papers. The conventions are self-consistent within each chapter, and they are kept as uniform as possible between the chapters. Chapter 1 reviews the basics of the maximal supergravity theories in eleven and ten dimensions. The role of the S- and T-duality in string theory are described, and the U-duality conjecture is presented. The theory of Kaluza-Klein compactification is reviewed in Chapter 2. The dimensional reduction on an n-dimensional torus is done explicitly, in particular the symmetry properties of the moduli space are analyzed in relation to the U-duality. Compactification on Calabi-Yau threefolds and G 2 manifolds is also discussed in this chapter. Moreover, this chapter also contains a brief account of the topological subsectors of string and M-theory residing on Calabi- Yau threefolds and G 2 manifolds. Chapter 3 picks up where Chapter 1 ends and discusses the low energy effective actions of type II string theories beyond the supergravity level. It contains both perturbative and non-perturbative quantum corrections, organized as a double expansion in the Regge slope α and the string coupling constant g s. Both supersymmetry and U-duality turn out to be useful for finding the general structures of the correction terms. The g s expansion at each α -level is encoded by mathematical functions called automorphic forms. Some mathematical backgrounds of automorphic forms is introduced in Chapter 4. The non-holomorphic Eisenstein series based on the discrete group SL(2, Z) is worked through in detail as a guiding example. Various construction methods as well as the Fourier properties of it are presented. Generalization to Eisenstein series based on discrete subgroups of larger Lie groups is discussed. Moreover, construction of automorphic forms which transform under certain Lie groups is briefly mentioned. One of the constructions is based on p-adic numbers, the relevant properties of this mathematical field are given in Appendix A. The appended research papers are grouped into two parts. The first part consists of Paper I and II and deals with the topological subsector of M- theory. The second part analyzes some symmetry structures of the quantum corrections in string theory effective actions. This part contains Paper III, 7

16 IV and V. Considering a target space with G 2 holonomy, the supersymmetric action of a membrane moving in this space is formulated in Paper I. The fact that this action is BRST-exact on-shell indicates that it is topological. It is suggested that this membrane is the fundamental object of the conjectured topological M-theory. The action for a five-brane in topological M-theory is subsequently given in Paper II using the top-form formulation. After compactification on a circle, the M5-brane is identified with the NS5-brane in the topological A model. The Kodaira-Spencer equation appears as equation of motion for the three-form on the NS5-brane, which indicates a duality relation between the topological A and B models. Paper III discusses symmetries of coset type for the gravitational R 2, R 3 and R 4 corrections in the string effective action. This is achieved by dimensional reduction to three spacetime dimensions on n-torii. It is argued in this paper that requiring invariance under U-duality would require transforming automorphic forms. The toroidal reduction of the Gauss-Bonnet combination is analyzed in detail in Paper IV. By investigating the dilaton exponents in the resulting action, the symmetry properties of this correction term are discussed. In particular focus is set on the U-duality symmetry SL(n +1, R). Paper V is also dealing with quantum corrections, although in another context. The system considered here is type IIA string theory compactified on a rigid Calabi-Yau threefold. The moduli space variables of this theory parameterizes the symmetric space SU(2, 1)/U(2). It is argued that the quantum corrections at the two-derivative level are captured by the non-holomorphic Eisenstein series based on the Picard modular group SU(2, 1; Z[i]). Physical interpretations are given for the various components of this Eisenstein series. 8

17 1 Supergravities and Dualities This chapter is devoted to the subject of supergravity theories, which initially were considered as candidates for the unification of the Standard Model with Einstein s theory of general relativity. Nowadays they are understood as low energy limits of string and M-theory. In the limit of large string tension, or equivalently, when the Regge slope α 0 the massive particles become very heavy. It is then justified to approximate string theory with its low-energy effective supergravity. Even though supergravity theories only describe interactions between the massless modes, studying them has proven to be very fruitful. Most importantly they opened the door to the powerful non-perturbative tools called dualities, resulting in the second superstring revolution. 1.1 Higher-Dimensional Supergravites A supergravity theory, originally proposed in Ref. [18], is the extension of gravity by supersymmetry. By definition it is invariant under local super-poincaré transformations. Among other fields supergravity contains a massless spintwo graviton and its superpartner, the spin 3/2 gravitino. Thenumberof gravitino fields is denoted by N and equals the number of copies of a supersymmetry. Supergravities can be formulated in many spacetime dimensions. However, constraining all the particle spins to be two or less, as is what has been observed in nature, it was shown in Ref. [19] that the maximal number of supercharges consistent with a single graviton is 32. This corresponds to an eleven-dimensional spacetime with Lorentzian metric 1. In this section we will concentrate on supergravities in D =10andD = 11, since they are most 1 Relaxing the Lorentzian metric constraint it is possible to have twelve dimensions with two of them being timelike, which is the background setup for the so called F-theory [20]. 9

18 closely related to string and M-theory. The standard reference to supersymmetry and supergravity is the book by J. Wess and J. Bagger [21] Eleven-Dimensional Supergravity Ever since its discovery [22] eleven-dimensional supergravity has held a special place in high energy theoretical physics. This is the only supersymmetric theory in eleven dimensions. It contains one supermultiplet, transforming as a single representation of the supergroup OSp(1 32). The field content of the supermultiplet consists of the elfbein EM A, the gravitino Ψ M and a rank three gauge field C MNP. The index M is the curved spacetime index, while A is its tangent space equivalent. Since it contains the maximal number of supersymmetries permitted in eleven dimensions, this theory is called a maximal supergravity. The gravitino is a 32-component Majorana spinor, which transforms as a representation under Spin(1, 10). The bosonic part of the eleven-dimensional supergravity action is S 11 = 1 [ d 11 x GR + 1 G 2κ 2 4 G ] G 4 G 4 C 3, (1.1) where R is the curvature scalar defined using the metric G MN = η AB EM A B EN. The four-form field strength G 4 dc 3 is invariant under the gauge transformations C 3 C 3 = C 3 + dλ 2 (1.2) and satisfies the Bianchi identity Einstein s equation together with dg 4 =0. (1.3) d G G 4 G 4 = 0 (1.4) constitute the equations of motion. An alternative formulation can be found by introducing also a dual gauge field C 6 and its corresponding field strength G 7 = dc C 3 G 4. (1.5) Requiring G 4 = G 7, (1.6) Eq. (1.4) turns into the Bianchi identity of G 7. The overall constant κ 11 is related to the eleven-dimensional Newton s constant G 11 and the 11-dimensional Planck length l p as 2κ 2 11 =16πG 11 = (2πl p) 9. (1.7) 2π 10

19 Using the supersymmetry variations A δem = εγa Ψ M, δc MNP = 3 εγ [MN Ψ P ], δψ M = M ε + 1 ( ! Γ MG NPQR Γ NPQR 1 ) 2 G MNPQΓ NPQ ε, (1.8) we can obtain the full supersymmetric action. The variation δψ M given here is only to leading order in fermionic fields, additional terms which are quadratic in the fermionic fields have been dropped. The Dirac matrices are defined by Γ M = EM AΓ A,withΓ A satisfying the Clifford algebra. Moreover, the covariant derivative appearing in Eq. (1.8) is given by M ε = M ε ω MABΓ AB ε, (1.9) where ω MAB is the standard spin connection in tangent space. In order to find bosonic solutions which also preserve some supersymmetries, the variation of the gravitino has to vanish: δψ M = M ε ( 1 4! Γ MG NPQR Γ NPQR 1 2 G MNPQΓ NPQ ) ε =0. (1.10) A spinor ε satisfying this equation is called a Killing spinor. More specifically for eleven-dimensional supergravity there are two stable maximally supersymmetric brane solutions, a 2-brane and a 5-brane, which are electrically and magnetically charged, respectively, with respect to the C 3 field. The fact that they both saturate the Bogomolny-Prasad-Sommerfield (BPS) bound means that their masses are equal to their charges. These two solutions are precisely the long-wavelength limits of the M2- and M5-brane in M-theory with T M2 =2π(2πl p ) 3 and T M5 =2π(2πl p ) 6 (1.11) being their tensions [4]. The uniqueness of eleven-dimensional supergravity caused much excitement when it was first introduced. Much of the hope of it being the Theory Of Everything died out when it was realized that D = 11 supergravity is non-chiral as well as non-renormalizable. However, it managed to come back to the forefront of physics when E. Witten pointed out the existence of eleven-dimensional M-theory. Instead of being a fundamental theory, D = 11 supergravity should be thought of as the classical limit of M-theory. The fact that it is not renormalizable is not an obstacle anymore since it is only an effective theory valid at low energies. Since then it has also been understood that four-dimensional chiral theories can be obtained from higher-dimensional non-chiral ones by compactifying on manifolds with suitable singularities [23, 24]. 11

20 1.1.2 Type IIA Supergravity The first hint towards M-theory is the construction of type IIA supergravity. This theory is obtained from eleven-dimensional supergravity by dimensional reduction [25]. Similar to how D = 11 supergravity is interpreted as the low energy limit of M-theory, type IIA supergravity is the low energy limit of type IIA superstring theory in ten dimensions [26]. Upon dimensional reduction on S 1, the eleven-dimensional metric gives rise to a ten-dimensional metric, a gauge field and a scalar (dilaton) in the following way ( ) gμν + e G MN = 2σ A μ A ν e 2σ A μ e 2σ A ν e 2σ. (1.12) The conventions used here are the same as in Ref. [27]. As opposed to compactification, in dimensional reduction only the zero modes in the Fourier expansions of the various fields are kept. Similarly the three-form gauge field is decomposed into a three-form and a two-form C μνρ = C μνρ (C νρ,10 A μ + cyclic), B μν = C μν,10. (1.13) The bosonic part of the dimensional reduced action can now be written as S IIA = 1 d 10 x [ ge σ R 1 2κ 2 2 4! F ! e 2σ H3 2 1 ] 4 e2σ F (1.14) B 4κ 2 2 dc 3 dc 3, where the field strengths are defined according to F 2 = da, H 3 = db 2, F 4 = dc 3 A H 3. (1.15) To bring the action to the standard string frame we need to rescale the metric g μν e σ g μν. The end result is S IIA,S = 1 2κ 2 d 10 x ( g [e 2φ R +4( φ) H2 3 ] 1 2 4! F F κ 2 B 2 dc 3 dc 3, ) (1.16) with φ 3σ/2. Later in Section 3.1 we will see that the factor e 2φ in front of the curvature scalar originates from a spherical string worldsheet. Sometimes it is useful to express the type IIA supergravity action without this dilaton factor, which is known as the Einstein frame. This can be achieved by yet 12

21 another Weyl rescaling of the metric, g μν e φ/2 g μν, yielding S IIA,E = 1 d 10 x g [(R 12 ) 2κ 2 ( φ)2 e φ 12 H2 3 ] eφ/2 2 4! F 4 2 e3φ/2 4 F B 4κ 2 2 dc 3 dc 3. (1.17) Notice that compared to Eq. (1.16) new couplings between the dilaton and the R-R fields have also appeared. Decomposing the gravitino in eleven dimensions into representations of Spin(1, 9), we obtain a Majorana gravitino (ψα μ ) and a Majorana dilatino (λ α ). Using the Γ 11 matrix each of these can be decomposed again into a pair of Majorana-Weyl spinors of opposite chirality. Together with the graviton (g μν ), the antisymmetric tensor (B μν ), the dilaton (φ), the vector (A μ ) and the antisymmetric three tensor (C μνρ ), they form a single supermultiplet of N =(1, 1) supersymmetry. All the supersymmetry transformations can be found in Ref. [25], in particular transformations of the fermionic fields in the Einstein frame are given by δλ = μφγ μ Γ 11 ε e3φ/4 F (2) μν Γμν ε + i eφ/4 F (4) μνρσγ μνρσ ε, δψ μ = μ ε e3φ/4 F νρ (2) (Γμνρ 14g μν Γ ρ )Γ 11 ε e φ/2 H νρσ (Γ μνρσ 9g μν Γ ρσ )Γ 11 ε + i 256 eφ/4 F (4) νρστ (Γ μνρστ 203 gμν Γ ρστ ) Γ 11 ε. i 24 2 e φ/2 H μνρ Γ μνρ ε (1.18) The covariant derivative is defined as μ ε = ( μ ωab μ Γ ab) ε. The full action of type IIA supergravity is obtained by acting on the bosonic part in Eq. (1.17) with supersymmetry transformations. The type IIA string coupling constant is defined in terms of the vacuum expectation value of the dilaton g s = e <φ>. (1.19) As a result of the dimensional reduction (1.12), the string length scale is related to the Planck constant via l p = gs 1/3 l s, (1.20) with l s = α. At the same time Newton s constant in ten and eleven dimensions are related as G 11 =2πR 11 G 10, (1.21) 13

22 where the radius of the compact circle is then found to be R 11 = g s l s.using 16πG 10 = 1 2π (2πl s) 8 (g s ) 2 (1.22) one can thus show that κ appearing in Eq. (1.17) should be defined as 2κ 2 = 1 2π (2πl s) 8. (1.23) Moreover, we find that the string coupling constant satisfies g s = ( R11 l p ) 3/2. (1.24) Just like the physical fields, most of the branes contained in IIA supergravity also have eleven-dimensional origins [28, 29]. The M2-brane wrapped on the compactified circle is a IIA fundamental string F1, with tension given by, in the string frame, T F1,S =2πR 11 T M2 = 1. (1.25) 2πl 2 s On the other hand, an M2-brane not wrapping around the compactified circle is a D2-brane. Similarly the M5-brane gives rise to a D4- or an NS5-brane. The origin of the D0- and D6-branes are slightly harder to guess. The former corresponds to the lowest Kaluza-Klein momentum mode along the compactified circle. The latter is the magnetic dual of the D0-brane, and its physical interpretation is a Kaluza-Klein monopole. The presence of a D8-brane would however lead to a mass deformation of the IIA supergravity. Since no elevendimensional lift of massive IIA supergravity is yet known, the origin of the D8-brane is not as well understood as the other branes. Once type IIA supergravity is formulated we can revert the argument. By going to the strong coupling limit, we would have rediscovered its elevendimensional origin [30, 29] Type IIB Supergravity Besides type IIA supergravity, there exists one more maximal supergravity in ten dimensions. This theory is called type IIB supergravity and describes the massless limit of the type IIB superstring [31, 32, 26]. The supermultiplet of type IIB supergravity contains the graviton (g μν ), two scalars (φ, C 0 ), two antisymmetric tensors (B 2,C 2 ), one self-dual four-form (C 4 ), two Majorana- Weyl gravitini of the same chirality (or one Weyl gravitino ψ μ ) and two Majorana-Weyl dilatini of the same chirality (or one Weyl dilatino λ). The metric, φ and B 2 belong to the NS-NS sector, while C 0, C 2 and C 4 belong 14

23 to the R-R sector. Since all the fermions are of the same chirality, type IIB supergravity is chiral and said to have N =(2, 0) supersymmetry. The property that the field strength of the four-form C 4 is self-dual is impossible to obtain from a covariant action. One can thus either work entirely at the level of equations of motion, or one can write down an action which yields all other equations except for the self-duality and then impose this condition by hand. It is the latter approach we are going to utilize 2. One important feature of this theory is the existence of a global SL(2, R) invariance. Elements of this matrix group { ( ) } a b SL(2, R) = γ = ad bc =1;a, b, c, d R (1.26) c d act by fractional linear transformations on the scalars aτ + b τ cτ + d, with τ = C 0 + ie φ, (1.27) and linearly on the two-forms ( ) ( )( ) B2 d c B2. b a (1.28) where C 2 The bosonic part of the action in the Einstein frame can now be written as S IIB,E = 1 d 10 x [ g R 1 τ τ 2κ 2 2 I(τ) G ] 2 5! F (1.29) C 4 G 3 2i G 3, H 3 = db 2, F 3 = dc 2, G 3 = i F 3 + τh 3 I(τ), F 5 = dc 4 C 2 H 3. (1.30) C 2 The notation I(x) is referring to the imaginary part of x. In addition we have to impose the self-duality condition F 5 = F 5. (1.31) Straightforward computation shows that both the action and the self-duality condition are invariant under SL(2, R). The choice of Einstein frame has made this quite transparent. In fact the invariance of the scalar sector can be made 2 There is also a manifestly covariant formulation, by extending the theory with an auxiliary scalar field together with an extra gauge symmetry [33]. 15

24 manifest once one observes that the moduli space, parameterized by φ and C 0, is isomorphic to the symmetric space SL(2, R)/SO(2). More about this matter later when we discuss S-duality. The supersymmetry transformations can be found in [31, 32], for instance the transformations of the fermions are give by δλ = 1 2 eφ Γ μ ε μ τ i 24 eφ/2 Γ μνρ εg (3) μνρ, δψ μ = μ ε i 1920 Γμ 1...μ 5 Γ μ εf μ (5) 1...μ (1.32) 96 (Γμνρσ 9g μν Γ ρσ ) ε G (3) νρσ, where ε = ε L + iε R and ε = ε L iε R define the complexified version of a left and a right Majorana-Weyl spinor. Acting recursively on the bosonic action with the supersymmetry transformations we will find the full supersymmetric action. The detailed computation can be found in Refs. [31, 32]. The brane content of IIB supergravity includes odd-dimensional D( 1)-, D1-, D3-, D5- and D7-branes, which act as sources for the R-R gauge fields. The D( 1)- and D7-branes are coupled electrically and magnetically to the C 0 potential, respectively. Similarly C 2 is coupled to D1 and D5, while C 4 is coupled to the self-dual D3. In addition there are electric and magnetic sources for the B 2 field, namely the fundamental string F1 and the NS5-brane, respectively The Democratic Formulation By extending the R-R fields with their Hodge duals, the authors of Ref. [34] managed to formulate both type IIA and IIB supergravity in a uniform way. The field content in this formulation becomes IIA : IIB : {g μν,b μν,φ,c (1),C (3),C (5),C (7),C (9),ψ μ,λ}, {g μν,b μν,φ,c (0),C (2),C (4),C (6),C (8),ψ μ,λ}. (1.33) The extra degrees of freedom will later be removed by self-duality constraints. Type IIA contains fermions of both chiralities, while the opposite is valid for IIB with Γ 11 ψ μ = ψ μ and Γ 11 λ = λ. The notations will be hugely simplified if we define a collective gauge potential C = 5, 9 2 n=1, 1 2 C (2n 1), (1.34) where the sums run over the integers in IIA and half-integers The field strengths are then given by in IIB. H = db and G = dc H C + G (0) e B, (1.35) 16

25 with G = 5, 9 2 G (2n) being a collective field strength. G (0) is the constant n=0, 1 2 mass parameter of IIA supergravity, while it vanishes in IIB supergravity. Notice that Eq. (1.35) should be read off order by order in form degrees, in particular the last term in G is only present in IIB. The corresponding Bianchi identities are given by dh =0 and dg = H G. (1.36) We are now ready to present the bosonic action: S = 1 d 10 x ( g e 2φ R 4( φ) ) 2κ 2 2 H H , 9 2 n=0, 1 2 G (2n) G (2n). (1.37) As already mentioned, to remove the extra degrees of freedom the gauge potentials have to obey self-duality constraints G (2n) =( 1) [n] G (10 2n) +(fermi), (1.38) where [n] refers the integer part of n. This formulation can also be applied to massive type IIA supergravity, where one also adds a nine-form C (9). The dual of its field strength satisfies G (0) = m, with m being the Romans mass. This theory was first constructed in Ref. [35] as a deformed version of ordinary IIA supergravity. Though its classical eleven-dimensional lift is so far not known, the theory should be containedinm-theory. As a side comment, a similar idea of grouping even and odd differential forms has also been employed in the context of generalized complex structures [36], although the reason behind it is of another character. There the geometry of a manifold is described by differential forms, with even and odd forms being mapped to Weyl spinors of different chiralities. 1.2 S-duality The SL(2, R) invariance of type IIB supergravity expressed in Einstein frame is a perfect example of a phenomenon known as S-duality. Since the coupling constant in that theory is defined as g s = e <φ>, physically the operation τ with τ being defined as 1 τ τ = C 0 + ie φ (1.39) corresponds to an inversion of the coupling constant. In other words, strong coupling physics maps to the weak coupling regime. 17

26 This kind of duality was first discovered as a duality between electric and magnetic quantities in Maxwell s equations. Later it was generalized to N = 4 super Yang-Mills theory under the name of Montonen-Olive duality [37]. The most general Lagrangian of N = 4 SYM has the following form L SYM = 1 Tr(F F )+ θ Tr(F F ). (1.40) g2 8π2 The second term is topological, and thus does not have any significance for the classical equations of motion. However, after quantization the story changes, since now the quantum states are characterized also by the θ angle. Furthermore, defining a modular parameter as τ = θ 4π + i, (1.41) 2π gym 2 the quantized theory is invariant under the modular group SL(2, Z) byfrac- tional transformations [38]. Similar behavior has also been studied in N =2 Seiberg-Witten gauge theories [39, 40]. Similar to the super Yang-Mills theory, quantizing IIB supergravity will break the continuous SL(2, R) to a symmetry of the modular group SL(2, Z) [41]. An intuitive understanding of this can be achieved by studying the BPS states in the theory. A BPS state is supersymmetric and saturates certain equality relations between its mass and charges. If the maximal number of supersymmetries are preserved we simply call it BPS, if only half of the supersymmetries are preserved we call it half-bps, etc.. One property that makes BPS states interesting is that they are protected by supersymmetry. That is, as long as the supersymmetry is unbroken they are stable under rescaling of the coupling constant, leading to many scaling independent properties. The only occasion this fails is when another representation becomes degenerate with the BPS multiplet, then a mechanism similar to the Higgs mechanism might take place. ThefactthattheSL(2, R) symmetry rotates the doublet (B 2,C 2 )makes the states coupled to these potentials suitable for study. It turns out that one can form a bound state of p F-strings and q D-strings. By the Dirac quantization argument the tensions of these so called (p, q) strings must take discrete values. Unlike the gauge potentials, the tensions are rotated by the discrete SL(2, Z) group. Starting from the tension of a fundamental string we can thus find the tension of an arbitrary (p, q) string by modular transformations, which in the Einstein frame is given by T (p,q) = p + qτ I(τ) T F1,S, p and q co-prime. (1.42) 18

27 Here T F1,S = 1 denotes the tension of a fundamental string in the string 2πls 2 frame. The F- and D-strings correspond precisely to the special cases (1, 0) and (0, 1), respectively: T F1,E = g s T F1,S, T D1,E = 1 T F1,S. (1.43) gs Notice that the D-string tension formula is valid only when R(τ) = 0. Since both F1 and D1 are 1/2-BPS states, the formula (1.42) is indeed valid for all couplings. At weak string coupling (g s 1), the D-strings are too heavy to be observed. The situation becomes the opposite at strong coupling. The S-duality, which manifests itself as the modular group, exchanges the roles of F- and D-strings. Lastly, a junction of three (p, q) strings requires charge conservation: i p(i) = i q(i) =0fori =1, 2, 3. Under the modular group, the D3-brane transforms as a singlet. Therefore S-duality does not pose any additional constrain on how (p, q) strings can end on a D3-brane. The D5- and NS5-branes can also be grouped into a stable (p, q) five-brane, which is the magnetic dual of the (p, q) string. The fluctuations of the D5-brane are described by F-strings attached to it, with the same relation being true also for NS5-brane and D-strings. The (p, q) five-brane has similar modular properties as the (p, q) string. The SL(2, Z) transformations on the D7-branes are however more complicated. In order to understand the S-duality at a deeper level we need first to introduce a new concept called T-duality. 1.3 T-duality T-duality is a symmetry of string theory which arises as a consequence of compactification on an n-torus T n. Before stating the symmetry group in the general case, let us first illustrate the phenomenon using the simplest example, the bosonic closed string compactified on a circle with radius R. The notion of circular compactification simply means that the string worldsheet along the compactified direction in the target space should have a periodic boundary condition X 25 (τ,σ + π) =X 25 (τ,σ)+2πwr, w Z, (1.44) where we have assumed the 25th space direction to be compact. The remaining spacetime coordinates are assumed for simplicity to be Minkowski. Here τ and σ are the standard worldsheet parameters. The discrete number w, called the winding number, denotes the number of times the string winds around the compact direction. The oscillator expansion in the compact direction then becomes X 25 (τ,σ)=x 25 +2α p 25 τ +2wRσ +(oscillators). (1.45) 19

28 Since X 25 is compact, the momentum of the center of mass along this direction, p 25, must be quantized p 25 = n, n Z. (1.46) R Dividing the expansion into left- and right-movers we may define X 25 (τ,σ)=x 25 L (τ + σ)+x25(τ σ), (1.47) P L = n α R + wr and P R = n α wr. (1.48) R It is now apparent that the mass squared [4] [ ( n ) ( ) ] 2 2 wr α M 2 = α + +2N R α L +2N R 4 (1.49) as well as the oscillator number matching condition are invariant under the transformation R N L N R = nw (1.50) R α, n w. (1.51) R The compact coordinate itself will transform as X 25 L X 25 L and X 25 R X 25 R, (1.52) and similar results are obtained for the respective currents. Not only the spectrum matches perfectly, also the interactions respect this so called T-duality. What T-duality really implies is that string theory compactified on a circle with radius R is equivalent to compactification on another circle with radius α /R, provided that the winding number and the momentum are interchanged at the same time. Note that the fact that the string can wind around the compact dimension is crucial for this duality to exist, and thus T-duality can never be a property of a compactified point-particle theory. The duality transformation in Eq. (1.51) is not a coincidence, and the reason can be understood as follows. The pair (P L,P R ) from Eq. (1.48) can be considered as vectors in a space endowed with the metric natural choice for the basis vectors of this space is ( α e 1 =(R, R) and e 2 = 20 R, α R ( 1 2α α ).A ), (1.53)

29 resulting in the following metric of scalar products ( ) ( ) e1 e ξ = 1 e 1 e =. (1.54) e 2 e 1 e 2 e Since the vectors (P L,P R ) are discrete quantities, they define an integer lattice with Lorentzian metric. Being the unique two-dimensional Lorentzian lattice which is also even and unimodular, this lattice is known in the literature as Π 1;1 : Π 1;1 = { m i e i m i Z; i =1, 2 }, (1.55) see Refs. [42, 43]. The symmetry group that leaves this lattice invariant is O(1, 1; Z) = { x GL(2, Z) x T ξx = ξ }. (1.56) Explicitly solving the equation x T ξx = ξ shows that O(1, 1; Z) = Z 2, where the only non-trivial solution precisely correspond to the exchange of momentum and winding number. Generalization to compactification of the superstring on an arbitrary ntorus T n is straightforward. The momenta and winding numbers then describe the even self-dual lattice Π 1;1... Π }{{ 1;1. (1.57) } n times The symmetry group of this lattice is the infinite discrete group O(n, n; Z), whichisdefinedby O(n, n; Z) = { x GL(2n, Z) x T ξx = ξ } (1.58) with ( ) 0 1n ξ = (1.59) 1 n 0 being the invariant metric. This is the most general result of T-duality [44]. One thing worth noticing is that T-duality, seen as a symmetry of M-theory, only acts on momentum excitations and winding modes, in other words, it is a perturbative symmetry. For instance the one-loop partition function has been shown to respect T-duality [44]. More generally, it is valid order by order in the g s expansion. Extending to open strings, it can be shown that under T-duality Neumann boundary conditions turn into Dirichlet ones. This property naturally lead to the concept Dirichlet-branes or D-branes, which are defined as hypersurfaces on which an open string can end. The D-branes themselves can then also undergo T-dualization. Later in the context of Calabi-Yau compactification a special type of T-duality has been extensively studied under the name mirror symmetry [45]. 21

30 1.4 U-duality Now we are fully equipped to return to the SL(2, Z) symmetry in IIB superstring theory. Compactifying the IIB theory on a circle with radius R B shows that it is actually equivalent to the IIA theory compactified on a circle with radius R A = α /R B. The fact that IIA and IIB theories are related via a T- duality indicates that they have a common higher-dimensional origin. One can now go on comparing IIB compactified on a circle with M-theory compactified on a two-torus. Study of BPS states, i.e., (p, q) string, D-branes, M-branes, etc., shows the matching again works perfectly well. Refs. [4, 27] provide some flavors of the kind of computation involved. The most interesting observation is, however, the identification τ M = τ B, (1.60) where τ M is the complex structure modulus of the two-torus on which M- theory is compactified, and τ B is the complex scalar of IIB theory defined in Eq. (1.26). This relation tells us the weak-strong coupling symmetry in IIB, which manifests itself as SL(2, Z) acting on τ B, can be interpreted as modular transformations on the M-theory two-torus [46, 47, 48]. S-duality has now received a geometric explanation! Though the relations are proven after compactification to nine dimensions, the symmetry should work even in the decompactification limit R B. We have just glimpsed at the interplay between S- and T-duality, both of them being symmetries of M-theory. Compactify now string theory on an (n 1)-torus. As already been argued in Section 1.3, the perturbative T-duality symmetry group becomes O(n 1,n 1; Z). When the determinant is 1 the T-transformation maps IIA and IIB to themselves, while if the determinant is 1 the T-transformation maps IIA IIB. This indicates that IIA and IIB are different sectors of one common underlying theory the eleven-dimensional M-theory. We can also switch to the M-theory point of view, where instead compactification on an n-torus has been performed. This leads to the S-duality group SL(n, Z), which is part of the diffeomorphism group yielding conformally equivalent n-torii. Together, S- and T-duality intertwine in a non-trivial way to generate the so called U-duality. All the U-duality groups are summarized in Table 1.1. Curiously they all belong to the E-series of exceptional Lie groups [49]. The first hint towards U-duality came from toroidal compactification of eleven-dimensional supergravity. Due to the simple geometry of T n, toroidal compactification preserves all supersymmetries, therefore the resulting supergravity theories are all maximal supersymmetric. By studying the scalar sector E. Cremmer and B. Julia were able to generalize the coset construction of the IIB moduli space [50]. They showed that in (11 n) dimensions the scalars of the compactified supergravity parameterize the symmetric space 22